eduKateSG · Judgement · The Gold Standard of Education
Use MOORA and MULTIMOORA as transparent comparisons, not automatic verdicts
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Full chapter index · Independent practice · Judgement subject guide
MOORA compares alternatives across several objectives after converting measurements to dimensionless ratios. Its ratio system aggregates benefit and cost objectives, while its reference-point form asks how far an alternative lies from desirable coordinates. MULTIMOORA adds a full multiplicative form and interprets agreement through dominance. The methods are compact, but their rankings still depend on objectives, directions, data quality, weights and the option set.
This guide builds a fictional three-option decision, preserves the raw matrix and tests what happens when a criterion direction, weight or near-zero value changes. The goal is a decision aid that a reader can recompute and challenge.
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Full chapter index
See the mechanism · Chapters 1–2
Build the worked model · Chapters 3–5
Test what changes · Chapters 6–8
Check evidence and limits · Chapters 9–10
Practise and continue · Chapters 11–12
Name the decision owner, alternatives, objectives, constraints and decision date. Separate hard requirements from compensable objectives. If an option violates a safety threshold, do not let a high convenience score compensate unless the policy explicitly permits that trade-off.
Define each objective’s unit, evidence source and direction. “Reliability” needs an observable measure; “cost” needs a time horizon. Record missing values and uncertainty before normalisation.
Evidence check: for Frame the decision before calculating, preserve the observation before the interpretation. Record decision owner, alternatives and constraints. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
Rows represent alternatives and columns represent objectives. Keep the raw matrix beside every transformed version. Label benefit objectives to maximise and cost objectives to minimise.
Imagine Options A, B and C with access scores 8, 6 and 9; reliability scores 7, 9 and 6; and weekly costs 6, 5 and 8 units. Use weights 0.40, 0.35 and 0.25 only after documenting how those trade-offs were elicited.
Evidence check: for Build the decision matrix, preserve the observation before the interpretation. Record objectives, directions and raw units. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
For each objective, divide each value by the square root of the sum of squared values in its column. Access therefore has denominator square root of 8 squared plus 6 squared plus 9 squared, about 13.45. The normalised access values are about 0.595, 0.446 and 0.669.
This transformation removes units but not measurement error or contested meaning. A score of 8 and a cost of 6 become comparable numerically only within the declared model.
Evidence check: for Normalise by vector length, preserve the observation before the interpretation. Record vector normalisation and assumptions. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
In an unweighted ratio system, add normalised benefit values and subtract normalised cost values. In a weighted version, multiply each normalised value by its declared weight first. Higher totals are preferred.
For Option A, the weighted contribution is 0.40 times normalised access plus 0.35 times normalised reliability minus 0.25 times normalised cost. Show each term, not only the total. That audit trail reveals sign errors immediately.
| Option | Access benefit | Reliability benefit | Weekly cost |
|---|---|---|---|
| A | 8 | 7 | 6 |
| B | 6 | 9 | 5 |
| C | 9 | 6 | 8 |
| Weight | 0.40 | 0.35 | 0.25 |
Evidence check: for Compute the ratio system, preserve the observation before the interpretation. Record every matrix and transformed term. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
Choose desirable coordinates from the best observed normalised value for each benefit and the lowest for each cost, unless a policy target supplies an external aspiration. Compute each alternative’s weighted absolute deviation from each coordinate and use the maximum deviation under the Tchebycheff logic.
The preferred alternative minimises its worst shortfall. This asks a different question from the additive ratio system. An option with strong average performance may lose because one objective is far from the reference.
Evidence check: for Construct the reference point, preserve the observation before the interpretation. Record ratio-system arithmetic. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
The full multiplicative form compares the product of benefit terms with the product of cost terms. Because products are sensitive to zero, negative values and scale conventions, inspect every transformed entry. Objectives must have meaningful non-negative treatment under the chosen formulation.
A tiny term can dominate the product. Use logarithms for stable computation when appropriate, but preserve the mathematical meaning. Do not shift negative data merely to make the formula run without explaining the changed preference model.
Evidence check: for Add the full multiplicative form carefully, preserve the observation before the interpretation. Record reference-point maximum deviation. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
MOORA’s ratio system, reference-point approach and full multiplicative form can agree or disagree. MULTIMOORA commonly interprets their ordinal outcomes through dominance rather than averaging incomparable scores.
Agreement across all three strengthens procedural robustness within this model. Disagreement is diagnostic. It shows that compensation, worst-deviation control and multiplicative balance favour different alternatives.
Evidence check: for Interpret the three rankings, preserve the observation before the interpretation. Record full multiplicative form and positivity. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
Suppose the ratio system ranks A first, C second and B third; the reference point ranks B first; and the multiplicative form ranks A first. A dominates two of three rankings but B exposes a weaker worst-case shortfall.
Do not hide that tension. Present A as the stronger balanced option under two forms and B as the reference-point safeguard. The decision owner can then decide whether average trade-offs or worst shortfalls better match the mandate.
Evidence check: for Work through a ranking disagreement, preserve the observation before the interpretation. Record dominance across the three forms. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
CHAPTER 9 OF 12 · Check evidence and limits
9. Test weights, directions and option-set dependence
Vary weights across plausible scenarios and note where first place changes. Revisit whether cost is truly a criterion to minimise or a constrained budget. Add measurement intervals and recompute rankings at their bounds.
Then add a clearly dominated option and check whether normalisation changes the ordering of existing alternatives. Because column denominators depend on all rows, the option set can affect scores. A rank change deserves disclosure.
Use a rank-stability map rather than a single alternative scenario. Vary two important weights across a grid while keeping their sum and the remaining weights coherent. Mark the regions where A, B or C ranks first under each MOORA form. A narrow winning region signals a fragile recommendation even when the baseline arithmetic is correct.
Measurement sensitivity should preserve correlations. If higher reliability estimates also tend to imply higher cost, changing those columns independently can create impossible scenarios. Use observed joint ranges or structured scenarios and record which combinations are plausible.
| Test | Change | Held fixed | Signal |
|---|---|---|---|
| Weight scenario | Trade-offs | Raw matrix | Preference sensitivity |
| Data interval | Measurements | Objectives | Evidence sensitivity |
| Dominated addition | Option set | Old rows | Normalisation effect |
| Veto threshold | Feasibility rule | Scores | Non-compensable risk |
Evidence check: for Test weights, directions and option-set dependence, preserve the observation before the interpretation. Record weight, interval and option-set sensitivity. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
MOORA does not elicit objectives or prove weights. Vector normalisation can mask the intuitive scale of trade-offs. The reference point may use observed extremes that are not substantively ideal. Products can be unstable around zeros.
A precise ranking is not precise evidence. If measurements are weak or stakeholders disagree about values, present scenarios, intervals and constraints rather than one authoritative number.
Do not use the three forms as three independent votes. They reuse the same matrix, so agreement is not independent statistical replication. It shows that the ordering survives three transformation logics. The shared data, objective set and weights can still be wrong.
Qualitative objectives need careful scales. Turning “community fit” into 1–10 values creates apparent ratios that may not be meaningful. Define anchor descriptions, use multiple assessors where appropriate and test adjacent ratings. If the scale supports only order, a cardinal arithmetic method may be unsuitable.
Document rounding only for display. Calculations should retain sufficient precision, because early rounding can create artificial ties or reverse near-equal alternatives. When final scores are close relative to measurement uncertainty, report a practically equivalent set instead of forcing a first-place label. The decision record should distinguish mathematical order from meaningful superiority.
Missing data should not be replaced by the most convenient neutral score. Explain the imputation model, test optimistic and pessimistic bounds, and show whether missingness is concentrated in one alternative. If a key objective is unmeasured, postponing the ranking may be more defensible than producing a complete-looking matrix from guesses.
Auditability also requires stable identifiers for alternatives and criteria. Store the raw source, transformation, direction, weight and timestamp for every cell. A later update can then be compared with the earlier decision without silently changing definitions.
Separate model validation from decision validation. Model validation checks arithmetic, formulas, units and reproducibility. Decision validation asks whether the objectives represent the mandate, affected people were heard, constraints are lawful and the result remains usable. A flawless spreadsheet can still answer the wrong question. Have an independent reviewer reproduce the matrix and a decision owner sign the interpretation, including disagreements among the three forms.
When alternatives evolve during deliberation, version them rather than overwriting rows. An improved Option B is a new decision object with a new evidence date. This preserves why the earlier ranking changed and prevents retrospective editing from making the process look more stable than it was.
Evidence check: for Know the method's limits, preserve the observation before the interpretation. Record compensation and method limits. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
Return to the matrix. Increase B’s weekly cost from 5 to 9 and recompute the cost denominator before every score. Predict which forms penalise the change most. Next, halve the reliability weight and redistribute the remainder to access.
Finally, declare reliability below 7 a hard veto. Option C now becomes infeasible even if an aggregate score is high. Explain why this is a governance rule, not another weight.
For independent practice, create a fourth option that exactly matches A except for lower cost. Confirm that it dominates A, then observe how adding it changes the column denominators and the scores of B and C. The exercise separates dominance in the raw objectives from changes caused by normalisation.
Next set one benefit value to zero. Explain what happens to the multiplicative form and why an arbitrary positive shift changes ratios. Propose a defensible alternative: revisit the measurement scale, treat zero as a boundary, or report only the forms whose assumptions are satisfied.
Evidence check: for Transfer check: change one assumption, preserve the observation before the interpretation. Record the changed-cost and veto contexts. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.
Is MOORA objective? Its arithmetic is reproducible, but objectives, data and weights remain choices. Does MULTIMOORA average three scores? No; the forms have different scales and are normally compared ordinally through dominance. Is the highest score automatically the decision? No; feasibility, uncertainty and accountability remain outside the arithmetic.
Primary reading includes Brauers and Zavadskas’s original 2006 paper, available through EuDML, and their full paper at Biblioteka Nauki. For the addition of the multiplicative form, see the technical discussion in Informatica. Continue with eduKate’s verified guides to COPRAS and WASPAS and lambda sensitivity.
Evidence check: for FAQs, routes and authoritative sources, preserve the observation before the interpretation. Record the narrowest defensible MOORA recommendation. Then state the inference, one credible alternative, and the additional observation that would distinguish them. This compact audit keeps a fluent explanation tied to evidence that another reader can inspect.
Changed-context transfer: repeat this chapter’s decision after changing one consequential condition while holding the others fixed. Calculate or predict what should change, name the threshold or mechanism, and compare with the original case. If the conclusion survives, state the range tested; if it reverses, explain why. Transfer is demonstrated by the changed case, not asserted from the first example.
Practical decision record: write one sentence for what is known, one for what remains uncertain, and one for the next safe action. Include the evidence owner, the latest useful decision time, and the condition that would stop or reverse the action. This turns a technical result into a reviewable choice without pretending uncertainty has disappeared.